arXiv subjects

Qian Lin

Publications and source records attributed to Qian Lin.

At least 37 records · Page 2Linked to original sources

arXiv2026

On the Eigenvalue Decay Rates of a Class of Neural-Network Related Kernel Functions Defined on General Domains

In this paper, we provide a strategy to determine the eigenvalue decay rate (EDR) of a large class of kernel functions defined on a general domain rather than $\mathbb S^{d}$. This class of kernel functions include but are not limited to the neural tangent kernel associated with neural networks with different depths and various activation functions. After proving that the dynamics of training the wide neural networks uniformly approximated that of the neural tangent kernel regression on general domains, we can further illustrate the minimax optimality of the wide neural network provided that the underground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, an interpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ of NTK. We also showed that the overfitted neural network can not generalize well. We believe our approach for determining the EDR of kernels might be also of independent interests.

stat.ML↗

arXiv2026

Supporting Evidence for the Adaptive Feature Program across Diverse Models

Theoretically exploring the advantages of neural networks might be one of the most challenging problems in the AI era. An adaptive feature program has recently been proposed to analyze feature learning, the characteristic property of neural networks, in a more abstract way. Motivated by the celebrated Le Cam equivalence, we advocate the over-parameterized sequence models to further simplify the analysis of the training dynamics of adaptive feature program and present several pieces of supporting evidence for the adaptive feature program. More precisely, after having introduced the feature error measure (FEM) to characterize the quality of the learned feature, we show that the FEM is decreasing during the training process of several concrete adaptive feature models including linear regression, single/multiple index models, etc. We believe that this hints at the potential successes of the adaptive feature program.

cs.LG↗

arXiv2025

Reg3D: Reconstructive Geometry Instruction Tuning for 3D Scene Understanding

The rapid development of Large Multimodal Models (LMMs) has led to remarkable progress in 2D visual understanding; however, extending these capabilities to 3D scene understanding remains a significant challenge. Existing approaches predominantly rely on text-only supervision, which fails to provide the geometric constraints required for learning robust 3D spatial representations. In this paper, we introduce Reg3D, a novel Reconstructive Geometry Instruction Tuning framework that addresses this limitation by incorporating geometry-aware supervision directly into the training process. Our key insight is that effective 3D understanding necessitates reconstructing underlying geometric structures rather than merely describing them. Unlike existing methods that inject 3D information solely at the input level, Reg3D adopts a dual-supervision paradigm that leverages 3D geometric information both as input and as explicit learning targets. Specifically, we design complementary object-level and frame-level reconstruction tasks within a dual-encoder architecture, enforcing geometric consistency to encourage the development of spatial reasoning capabilities. Extensive experiments on ScanQA, Scan2Cap, ScanRefer, and SQA3D demonstrate that Reg3D delivers substantial performance improvements, establishing a new training paradigm for spatially aware multimodal models.

cs.CV↗

arXiv2025

On the Pinsker bound of inner product kernel regression in large dimensions

Building on recent studies of large-dimensional kernel regression, particularly those involving inner product kernels on the sphere $\mathbb{S}^{d}$, we investigate the Pinsker bound for inner product kernel regression in such settings. Specifically, we address the scenario where the sample size $n$ is given by $αd^γ(1+o_{d}(1))$ for some $α, γ>0$. We have determined the exact minimax risk for kernel regression in this setting, not only identifying the minimax rate but also the exact constant, known as the Pinsker constant, associated with the excess risk.

math.ST↗

arXiv2025

On the Optimality of Functional Sliced Inverse Regression

In this paper, we prove that functional sliced inverse regression (FSIR) achieves the optimal (minimax) rate for estimating the central space in functional sufficient dimension reduction problems. First, we provide a concentration inequality for the FSIR estimator of the covariance of the conditional mean. Based on this inequality, we establish the root-$n$ consistency of the FSIR estimator of the image of covariance of the conditional mean. Second, we apply the most widely used truncated scheme to estimate the inverse of the covariance operator and identify the truncation parameter that ensures that FSIR can achieve the optimal minimax convergence rate for estimating the central space. Finally, we conduct simulations to demonstrate the optimal choice of truncation parameter and the estimation efficiency of FSIR. To the best of our knowledge, this is the first paper to rigorously prove the minimax optimality of FSIR in estimating the central space for multiple-index models and general $Y$ (not necessarily discrete).

math.ST↗

arXiv2025

Emotion Recognition Using Convolutional Neural Networks

Emotion has an important role in daily life, as it helps people better communicate with and understand each other more efficiently. Facial expressions can be classified into 7 categories: angry, disgust, fear, happy, neutral, sad and surprise. How to detect and recognize these seven emotions has become a popular topic in the past decade. In this paper, we develop an emotion recognition system that can apply emotion recognition on both still images and real-time videos by using deep learning. We build our own emotion recognition classification and regression system from scratch, which includes dataset collection, data preprocessing , model training and testing. Given a certain image or a real-time video, our system is able to show the classification and regression results for all of the 7 emotions. The proposed system is tested on 2 different datasets, and achieved an accuracy of over 80\%. Moreover, the result obtained from real-time testing proves the feasibility of implementing convolutional neural networks in real time to detect emotions accurately and efficiently.

cs.CV↗

arXiv2025

Neural Tangent Kernel of Neural Networks with Loss Informed by Differential Operators

Spectral bias is a significant phenomenon in neural network training and can be explained by neural tangent kernel (NTK) theory. In this work, we develop the NTK theory for deep neural networks with physics-informed loss, providing insights into the convergence of NTK during initialization and training, and revealing its explicit structure. We find that, in most cases, the differential operators in the loss function do not induce a faster eigenvalue decay rate and stronger spectral bias. Some experimental results are also presented to verify the theory.

cs.LG↗

arXiv2025

On the Saturation Effects of Spectral Algorithms in Large Dimensions

The saturation effects, which originally refer to the fact that kernel ridge regression (KRR) fails to achieve the information-theoretical lower bound when the regression function is over-smooth, have been observed for almost 20 years and were rigorously proved recently for kernel ridge regression and some other spectral algorithms over a fixed dimensional domain. The main focus of this paper is to explore the saturation effects for a large class of spectral algorithms (including the KRR, gradient descent, etc.) in large dimensional settings where $n \asymp d^γ$. More precisely, we first propose an improved minimax lower bound for the kernel regression problem in large dimensional settings and show that the gradient flow with early stopping strategy will result in an estimator achieving this lower bound (up to a logarithmic factor). Similar to the results in KRR, we can further determine the exact convergence rates (both upper and lower bounds) of a large class of (optimal tuned) spectral algorithms with different qualification $τ$'s. In particular, we find that these exact rate curves (varying along $γ$) exhibit the periodic plateau behavior and the polynomial approximation barrier. Consequently, we can fully depict the saturation effects of the spectral algorithms and reveal a new phenomenon in large dimensional settings (i.e., the saturation effect occurs in large dimensional setting as long as the source condition $s>τ$ while it occurs in fixed dimensional setting as long as $s>2τ$).

stat.ML↗

arXiv2025

Diagonal Over-parameterization in Reproducing Kernel Hilbert Spaces as an Adaptive Feature Model: Generalization and Adaptivity

This paper introduces a diagonal adaptive kernel model that dynamically learns kernel eigenvalues and output coefficients simultaneously during training. Unlike fixed-kernel methods tied to the neural tangent kernel theory, the diagonal adaptive kernel model adapts to the structure of the truth function, significantly improving generalization over fixed-kernel methods, especially when the initial kernel is misaligned with the target. Moreover, we show that the adaptivity comes from learning the right eigenvalues during training, showing a feature learning behavior. By extending to deeper parameterization, we further show how extra depth enhances adaptability and generalization. This study combines the insights from feature learning and implicit regularization and provides new perspective into the adaptivity and generalization potential of neural networks beyond the kernel regime.

cs.LG↗

arXiv2024

Offline Multi-Agent Reinforcement Learning via In-Sample Sequential Policy Optimization

Offline Multi-Agent Reinforcement Learning (MARL) is an emerging field that aims to learn optimal multi-agent policies from pre-collected datasets. Compared to single-agent case, multi-agent setting involves a large joint state-action space and coupled behaviors of multiple agents, which bring extra complexity to offline policy optimization. In this work, we revisit the existing offline MARL methods and show that in certain scenarios they can be problematic, leading to uncoordinated behaviors and out-of-distribution (OOD) joint actions. To address these issues, we propose a new offline MARL algorithm, named In-Sample Sequential Policy Optimization (InSPO). InSPO sequentially updates each agent's policy in an in-sample manner, which not only avoids selecting OOD joint actions but also carefully considers teammates' updated policies to enhance coordination. Additionally, by thoroughly exploring low-probability actions in the behavior policy, InSPO can well address the issue of premature convergence to sub-optimal solutions. Theoretically, we prove InSPO guarantees monotonic policy improvement and converges to quantal response equilibrium (QRE). Experimental results demonstrate the effectiveness of our method compared to current state-of-the-art offline MARL methods.

cs.AI↗

arXiv2024

Erbium doped yttrium oxide thin films grown by chemical vapour deposition for quantum technologies

The obtention of quantum-grade rare-earth doped oxide thin films that can be integrated with optical cavities and microwave resonators is of great interest for the development of scalable quantum devices. Among the different growth methods, Chemical Vapour Deposition (CVD) offers high flexibility and has demonstrated the ability to produce oxide films hosting rare-earth ions with narrow linewidths. However, growing epitaxial films directly on silicon is challenging by CVD due to a native amorphous oxide layer formation at the interface. In this manuscript, we investigate the CVD growth of erbium-doped yttrium oxide (Er:Y2O3) thin films on different substrates, including silicon, sapphire, quartz or yttria stabilized zirconia (YSZ). Alternatively, growth was also attempted on an epitaxial Y2O3 template layer on Si (111) prepared by molecular beam epitaxy (MBE) in order to circumvent the issue of the amorphous interlayer. We found that the substrate impacts the film morphology and the crystalline orientations, with different textures observed for the CVD film on the MBE-oxide/Si template (111) and epitaxial growth on YSZ (001). In terms of optical properties, Er3+ ions exhibit visible and IR emission features that are comparable for all samples, indicating a high-quality local crystalline environment regardless of the substrate. Our approach opens interesting prospects to integrate such films into scalable devices for optical quantum technologies.

cond-mat.mtrl-sci↗

arXiv2024

Improving Adaptivity via Over-Parameterization in Sequence Models

It is well known that eigenfunctions of a kernel play a crucial role in kernel regression. Through several examples, we demonstrate that even with the same set of eigenfunctions, the order of these functions significantly impacts regression outcomes. Simplifying the model by diagonalizing the kernel, we introduce an over-parameterized gradient descent in the realm of sequence model to capture the effects of various orders of a fixed set of eigen-functions. This method is designed to explore the impact of varying eigenfunction orders. Our theoretical results show that the over-parameterization gradient flow can adapt to the underlying structure of the signal and significantly outperform the vanilla gradient flow method. Moreover, we also demonstrate that deeper over-parameterization can further enhance the generalization capability of the model. These results not only provide a new perspective on the benefits of over-parameterization and but also offer insights into the adaptivity and generalization potential of neural networks beyond the kernel regime.

cs.LG↗

arXiv2024

On the Impacts of the Random Initialization in the Neural Tangent Kernel Theory

This paper aims to discuss the impact of random initialization of neural networks in the neural tangent kernel (NTK) theory, which is ignored by most recent works in the NTK theory. It is well known that as the network's width tends to infinity, the neural network with random initialization converges to a Gaussian process $f^{\mathrm{GP}}$, which takes values in $L^{2}(\mathcal{X})$, where $\mathcal{X}$ is the domain of the data. In contrast, to adopt the traditional theory of kernel regression, most recent works introduced a special mirrored architecture and a mirrored (random) initialization to ensure the network's output is identically zero at initialization. Therefore, it remains a question whether the conventional setting and mirrored initialization would make wide neural networks exhibit different generalization capabilities. In this paper, we first show that the training dynamics of the gradient flow of neural networks with random initialization converge uniformly to that of the corresponding NTK regression with random initialization $f^{\mathrm{GP}}$. We then show that $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 1$ for any $s < \frac{3}{d+1}$ and $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 0$ for any $s \geq \frac{3}{d+1}$, where $[\mathcal{H}^{\mathrm{NT}}]^{s}$ is the real interpolation space of the RKHS $\mathcal{H}^{\mathrm{NT}}$ associated with the NTK. Consequently, the generalization error of the wide neural network trained by gradient descent is $Ω(n^{-\frac{3}{d+3}})$, and it still suffers from the curse of dimensionality. On one hand, the result highlights the benefits of mirror initialization. On the other hand, it implies that NTK theory may not fully explain the superior performance of neural networks.

stat.ML↗

arXiv2024

An Offline Adaptation Framework for Constrained Multi-Objective Reinforcement Learning

In recent years, significant progress has been made in multi-objective reinforcement learning (RL) research, which aims to balance multiple objectives by incorporating preferences for each objective. In most existing studies, specific preferences must be provided during deployment to indicate the desired policies explicitly. However, designing these preferences depends heavily on human prior knowledge, which is typically obtained through extensive observation of high-performing demonstrations with expected behaviors. In this work, we propose a simple yet effective offline adaptation framework for multi-objective RL problems without assuming handcrafted target preferences, but only given several demonstrations to implicitly indicate the preferences of expected policies. Additionally, we demonstrate that our framework can naturally be extended to meet constraints on safety-critical objectives by utilizing safe demonstrations, even when the safety thresholds are unknown. Empirical results on offline multi-objective and safe tasks demonstrate the capability of our framework to infer policies that align with real preferences while meeting the constraints implied by the provided demonstrations.

cs.LG↗

arXiv2024

On the Optimality of Misspecified Spectral Algorithms

In the misspecified spectral algorithms problem, researchers usually assume the underground true function $f_ρ^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_ρ^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > α_{0}$ where $α_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the spectral algorithms are optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that spectral algorithms are minimax optimal for any $α_{0}-\frac{1}β < s < 1$, where $β$ is the eigenvalue decay rate of $\mathcal{H}$. We also give several classes of RKHSs whose embedding index satisfies $ α_0 = \frac{1}β $. Thus, the spectral algorithms are minimax optimal for all $s\in (0,1)$ on these RKHSs.

math.ST↗

arXiv2024

On the Structural Dimension of Sliced Inverse Regression

In this work, we address the longstanding puzzle that Sliced Inverse Regression (SIR) often performs poorly for sufficient dimension reduction when the structural dimension $d$ (the dimension of the central space) exceeds 4. We first show that in the multiple index model $Y=f( \mathbf{P} \boldsymbol{X})+ε$ where $\boldsymbol{X}$ is a $p$-standard normal vector, $ε$ is an independent noise, and $\mathbf{P}$ is a projection operator from $\mathbb R^{p}$ to $\mathbb R^{d}$, if the link function $f$ follows the law of a Gaussian process, then with high probability, the $d$-th eigenvalue $λ_{d}$ of $\mathrm{Cov}\left[\mathbb{E}(\boldsymbol{X}\mid Y)\right]$ satisfies $λ_{d}\leq C e^{-θd}$ for some positive constants $C$ and $θ$. We then focus on the low signal regime where $λ_{d}$ can be arbitrarily small and not larger than $d^{-8.1}$, and prove that the minimax risk of estimating the central space is lower bounded by $\frac{dp}{nλ_{d}}$. Combining these two results, we provide a convincing explanation for the poor performance of SIR when $d$ is large, a phenomenon that has perplexed researchers for nearly three decades. The technical tools developed here may be of independent interest for studying other sufficient dimension reduction methods.

math.ST↗

arXiv2024

Optimal Rate of Kernel Regression in Large Dimensions

We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^γ$ for some $γ>0$ ). We first build a general tool to characterize the upper bound and the minimax lower bound of kernel regression for large dimensional data through the Mendelson complexity $\varepsilon_{n}^{2}$ and the metric entropy $\bar{\varepsilon}_{n}^{2}$ respectively. When the target function falls into the RKHS associated with a (general) inner product model defined on $\mathbb{S}^{d}$, we utilize the new tool to show that the minimax rate of the excess risk of kernel regression is $n^{-1/2}$ when $n\asymp d^γ$ for $γ=2, 4, 6, 8, \cdots$. We then further determine the optimal rate of the excess risk of kernel regression for all the $γ>0$ and find that the curve of optimal rate varying along $γ$ exhibits several new phenomena including the multiple descent behavior and the periodic plateau behavior. As an application, For the neural tangent kernel (NTK), we also provide a similar explicit description of the curve of optimal rate. As a direct corollary, we know these claims hold for wide neural networks as well.

stat.ML↗

arXiv2024

On the Saturation Effect of Kernel Ridge Regression

The saturation effect refers to the phenomenon that the kernel ridge regression (KRR) fails to achieve the information theoretical lower bound when the smoothness of the underground truth function exceeds certain level. The saturation effect has been widely observed in practices and a saturation lower bound of KRR has been conjectured for decades. In this paper, we provide a proof of this long-standing conjecture.

stat.ML↗